The Effective Ehrenpreis Conjecture

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_\epsilon \to M$, and $N_\epsilon \to N$, can be chosen to have the degrees less than $\epsilon^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other.